Why Is the Discriminant Important?

Why do we need to know the discriminant anyway? If we already have the quadratic formula to find the exact answers to a quadratic equation, why would we look at just part of the formula?

The discriminant gives us important information about the quadratic equation. First, it can tell us how many solutions the quadratic equation has. It can also tell us how many times the graph crosses the x-axis and if the solutions are real or complex.

But the discriminant is just a number! How does it give us all that information? Remember, the discriminant is the same as the number you get under the square root in the quadratic formula. The plus or minus in the quadratic formula is the part that often gives you two different solutions.

A Positive Discriminant

If the discriminant is positive, this means that you have a positive number under the square root in the quadratic formula. This means you will end up with 2 real solutions. You can always take the square root of a positive number. It might not come out to a whole number, but it's going to be a real number. The plus or minus in the quadratic formula means you'll have to add this number to get one answer and you'll subtract it to find the second answer.

If there are 2 real solutions, this also means that the graph will have 2 x-intercepts. Remember, the solutions to a quadratic equation are often called roots or zeros. The roots/zeros/solutions are the the values for x that make the equation equal to 0. On a graph, this will be where the parabola crosses the x-axis. Anytime the discriminant is positive, the graph will cross the x-axis twice.

The discriminant won't tell you the actual answers. It doesn't tell you exactly where the graph crosses the x-axis, but it can tell you how many solutions and how many times it crosses. If you want to know more specifics, you have to use the entire Quadratic Formula to find the specific answers.

The example below shows an example of a quadratic equation with a positive discriminant: